Compressive Beam-Pattern-Aware Near-field Beam Training via Total Variation Denoising

Extremely large antenna arrays envisioned for 6G incurs near-field effect, where steering vector depends on angles and range simultaneously. Polar-domain near-field codebooks can focus energy accurately but incur extra two-dimensional sweeping overhead; compressed-sensing (CS) approaches with Gaussian-masked DFT sensing offer a lower-overhead alternative. This letter revisits near-field beam training using conventional DFT codebooks. Unlike far-field responses that concentrate energy on a few isolated DFT beams, near-field responses produce contiguous, plateau-like energy segments with sharp transitions in the DFT beamspace. Pure LASSO denoising, therefore, tends to over-shrink magnitudes and fragment plateaus. We propose a beam-pattern-preserving beam training scheme for multiple-path scenarios that combines LASSO with a lightweight denoising pipeline: LASSO to suppress small-amplitude noise, followed by total variation (TV) to maintain plateau levels and edge sharpness. The two proximal steps require no near-field codebook design. Simulations with Gaussian pilots show consistent NMSE and cosine-similarity gains over least squares and LASSO at the same pilot budget.

💡 Research Summary

The paper addresses a pressing challenge in next‑generation (6G) wireless systems that employ extremely large antenna arrays (ELAA). When the user or scatterer lies within the radiative near‑field region, the array steering vector depends on both angle and range, deviating from the conventional far‑field plane‑wave model. Existing near‑field solutions typically rely on polar‑domain codebooks that jointly sample angle and distance. While accurate, such codebooks double the dimensionality of the search space, leading to prohibitive pilot overhead and storage requirements, and they are not compatible with current angle‑only CSI frameworks.

The authors propose a fundamentally different approach: retain the conventional discrete Fourier transform (DFT) codebook that is already standardized for far‑field operation, and exploit the fact that a near‑field path manifests in the DFT beamspace as a contiguous, plateau‑like region with sharp edges rather than an isolated peak. By analyzing the 2‑D uniform planar array (UPA) under a separable Fresnel approximation, they derive closed‑form expressions for the 6 dB lobe widths along the two spatial axes (u = sinθ sinφ and v = cosθ). These widths, B_y and B_z, quantify how far the energy of a single path spreads across neighboring DFT beams. The product B_y · B_z, divided by the DFT grid spacings, yields the number of active DFT coefficients per path. Even for a 128 × 16 UPA at 28 GHz, the expected number of non‑zero coefficients for a multi‑path channel (L = 5) is only about eight, confirming that the near‑field channel remains highly sparse in the DFT domain, but the sparsity is clustered rather than scattered.

Standard ℓ₁‑regularized least squares (LASSO) is known to promote isolated non‑zeros and to shrink coefficient magnitudes uniformly. When applied to the near‑field beamspace, LASSO tends to attenuate the plateau height and fragment the contiguous support, thereby destroying the very structure that could be leveraged for efficient beamforming. To remedy this, the authors introduce total variation (TV) regularization, a technique widely used in image processing to enforce piecewise‑constant solutions while preserving sharp edges. By combining LASSO with TV, the algorithm first suppresses low‑amplitude noise and then refines the remaining support to retain flat plateau levels and crisp boundaries.

The proposed training procedure consists of three stages:

1. Support detection via LASSO – The BS transmits M ≪ N pilot beams formed by masking the 2‑D DFT matrix with i.i.d. Gaussian entries. The received vector y = Φs + w (Φ = P F, where P contains the Gaussian masks) is processed with LASSO (parameter λ₁) to obtain a coarse estimate b_s^(ℓ₁). The largest k magnitudes are selected to form an initial support mask M₀.

2. 2‑D support dilation – Recognizing that a true near‑field lobe occupies a contiguous rectangle, the mask M₀ is dilated by a user‑defined radius (r_y, r_z) along the two axes, yielding an expanded mask f_M that is highly likely to cover the entire plateau.

3. Magnitude‑TV refinement – Within f_M, the algorithm solves a constrained optimization problem that minimizes the squared error plus λ_TV · TV(|S|), where S is the reshaped beamspace matrix and TV denotes the anisotropic 2‑D total variation of the magnitude map. Coefficients outside f_M are forced to zero. The solution provides a refined beamspace estimate b_S, which is vectorized back to b_s, transformed to the array domain via the inverse DFT, and finally normalized to obtain the beamforming vector.

Simulation results validate the concept. Using a 128 × 16 UPA, three to five multipath channels, and SNR ranging from 0 dB to 20 dB, the LASSO+TV method consistently outperforms pure LASSO and least‑squares (LS) baselines. NMSE improvements of roughly 2 dB are observed across the SNR range, and the normalized beamforming gain (cosine similarity) is higher by 5–10 % for the same pilot budget. Visualizations of the recovered beamspace clearly show that the plateau’s extent and height are accurately captured, whereas LASSO alone yields a fragmented, attenuated pattern.

Crucially, the approach requires no specialized near‑field codebook design; it can be implemented on top of existing DFT‑based codebooks, preserving compatibility with current standards and hardware. This low‑overhead, structure‑aware compressive training framework thus offers a practical pathway to near‑field beam alignment in future massive‑MIMO deployments. The authors suggest extensions to multi‑user MIMO, dynamic mobility scenarios, and adaptive TV parameter selection via learning‑based methods as promising directions for future work.